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NDA 2 2025 Maths, question 75

Topic: Differential calculus

Passage · Q73 to Q75

Consider the function f(x)=x∣x∣f(x)=x|x|.

Q75

Consider the following statements :

I. The function is increasing in the interval (−∞,∞)(-\infty ,\infty ).

II. The function is differentiable at x=0x=0.

Which of the statements given above is/are correct?

  1. (a) I only
  2. (b) II only
  3. (c) Both I and IICorrect answer
  4. (d) Neither I nor II

Solution.

Answer: (c) Both I and II

  1. For x<0, f=−x² and f′=−2x>0; for x>0, f=x² and f′=2x>0.
  2. At zero the derivative is lim[h|h|/h]=lim|h|=0.
  3. Values also increase across zero, so both statements are true.